Simplify (a + b)° a sin (180°) + ab sin (270°) + b sin (360°) cos (360°) + (a – b)´ csc (270°) -
Simplify (a + b)° a sin (180°) + ab sin (270°) + b sin (360°) cos (360°) + (a – b)´ csc (270°) -
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
Related questions
Question
![**Simplify the Expression**
\[
\frac{{(a + b)^2 \cos(360^\circ) + (a - b)^2 \csc(270^\circ)}}{{a \sin(180^\circ) + ab \sin(270^\circ) + b \sin(360^\circ)}}.
\]
**Steps to Simplify:**
1. **Trigonometric Identities:**
- \(\cos(360^\circ) = 1\)
- \(\csc(270^\circ) = -1\)
- \(\sin(180^\circ) = 0\)
- \(\sin(270^\circ) = -1\)
- \(\sin(360^\circ) = 0\)
2. **Substitute and Simplify:**
- Numerator:
\[
(a + b)^2 \cdot 1 + (a - b)^2 \cdot (-1)
= (a + b)^2 - (a - b)^2
\]
- Denominator:
\[
a \cdot 0 + ab \cdot (-1) + b \cdot 0
= -ab
\]
3. **Expanded Form:**
- Expand \((a + b)^2\) and \((a - b)^2\):
\[
(a + b)^2 = a^2 + 2ab + b^2
\]
\[
(a - b)^2 = a^2 - 2ab + b^2
\]
- Substitute into the numerator:
\[
(a^2 + 2ab + b^2) - (a^2 - 2ab + b^2) = 4ab
\]
4. **Final Result:**
\[
\frac{4ab}{-ab} = -4
\]
The simplified value of the given expression is \(-4\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6bd28eaf-fd57-4a31-9955-56d743d3276b%2F840079c0-7024-449a-82dc-9b868141f3f5%2Ficwwzaq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Simplify the Expression**
\[
\frac{{(a + b)^2 \cos(360^\circ) + (a - b)^2 \csc(270^\circ)}}{{a \sin(180^\circ) + ab \sin(270^\circ) + b \sin(360^\circ)}}.
\]
**Steps to Simplify:**
1. **Trigonometric Identities:**
- \(\cos(360^\circ) = 1\)
- \(\csc(270^\circ) = -1\)
- \(\sin(180^\circ) = 0\)
- \(\sin(270^\circ) = -1\)
- \(\sin(360^\circ) = 0\)
2. **Substitute and Simplify:**
- Numerator:
\[
(a + b)^2 \cdot 1 + (a - b)^2 \cdot (-1)
= (a + b)^2 - (a - b)^2
\]
- Denominator:
\[
a \cdot 0 + ab \cdot (-1) + b \cdot 0
= -ab
\]
3. **Expanded Form:**
- Expand \((a + b)^2\) and \((a - b)^2\):
\[
(a + b)^2 = a^2 + 2ab + b^2
\]
\[
(a - b)^2 = a^2 - 2ab + b^2
\]
- Substitute into the numerator:
\[
(a^2 + 2ab + b^2) - (a^2 - 2ab + b^2) = 4ab
\]
4. **Final Result:**
\[
\frac{4ab}{-ab} = -4
\]
The simplified value of the given expression is \(-4\).
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